Partition-folding bijection conjecture for Macdonald polynomials

Let $\mu=(\mu_1,\ldots,\mu_k,1^m)$ and let $\nu=(\mu_1,\ldots,\mu_{k-1},\mu_k+1,1^{m-1})$, where $m>0$ and $\mu_k<\mu_{k-1}$. Two permutations are $\mu$-equivalent when they lie in the same generalized dual equivalence class for $\mu$. The inverse descent set is the descent set of the inverse permutation. For a partition $\mu$, let $\varphi_\mu$ be the sequence of bijections taking $(1^n)$ to $\mu$, and let $\mathrm{SS}(\lambda)$ denote the relevant set of permutations associated with $\lambda$.

Partition-folding bijection conjecture. There exists a bijection $\varphi$ on permutations that preserves the inverse descent set such that, if $u$ and $v$ are $\mu$-equivalent, then $\varphi(u)$ and $\varphi(v)$ are $\nu$-equivalent. In particular, for any partition $\mu$,

$$\widetilde{H}_{\mu}(X;q,t)=\sum_{\lambda}\left(\sum_{u\in\mathrm{SS}(\lambda)}q^{\mathrm{inv}_{\mu}(\varphi_{\mu}(u))}t^{\mathrm{maj}_{\mu}(\varphi_{\mu}(u))}\right)s_{\lambda}(X),$$

where $\varphi_\mu$ is the unique sequence of bijections taking $(1^n)$ to $\mu$.

The conjecture strengthens the preceding generalized dual equivalence statement by proposing explicit bijections that fold partitions toward more general shapes. The source reports that it had been tested for partitions up to size $11$, but the supplied text gives no resolution status.